Question

write an algorithim for langrange forlumaa finf the polynomial which fits the following data (-1,7) (1,5) (2,15)​

Answer 1

Lagrange polynomials are used for polynomial interpolation. For a given set of distinct points xj and numbers yj. Lagrange’s interpolation is also an Nth degree polynomial approximation to f(x).

Find the Lagrange Interpolation Formula given below,

lagrange

Solved Examples

Question: Find the value of y at x = 0 given some set of values (-2, 5), (1, 7), (3, 11), (7, 34)?

Solution:

Given the known values are,

x = 0 ; x0 = -2 ; x1 = 1 ; x2= 3 ; x3 = 7 ; y0 = 5 ; y1 = 7 ; y2 = 11 ; y3 = 34

Using the interpolation formula,

y = (x−x1)(x−x2)…..(x−xn)(x0−x1)(x0−x2)…..(x0−xn) y0 + (x−x0)(x−x2)…..(x−xn)(x1−x0)(x1−x2)…..(x1−xn) y1 + …. + (x−x1)(x−x1)…..(x−xn−1)(xn−x0)(x0−x1)…..(xn−xn−1) yn

y = (0−1)(0−3)(0−7)(−2−1)(−2−3)(−2−7) × 5 + (0+2)(0−3)(0−7)(1+2)(1−3)(1−7) × 7 + (0+2)(0−1)(0−7)(3+2)(3−1)(3−7) × 11 + (0+2)(0−1)(0−3)(7+2)(7−1)(7−3) × 34

y = 2127 + 496 + −7720 + 5154

y = 1087180